Optimal Protected Area Design for Allee Effect Mitigation in Spatial Predator-Prey Systems

TL;DR

This paper models predator-prey dynamics with the Allee effect in a spatial setting, exploring how protected areas can be optimally designed to prevent extinction of low-density populations through simulation and optimization techniques.

Contribution

It introduces a bi-objective optimization framework for designing protected areas in predator-prey models with Allee effects, incorporating both simulation and mathematical analysis.

Findings

Optimal protected area configurations vary between fragmented and contiguous patterns.

Protected areas with positive growth terms can facilitate population survival.

The model extends to predator-prey systems, analyzing steady states and persistence.

Abstract

Endangered populations often experience limited growth ability at low densities, a phenomenon described by the Allee effect. In this thesis, we investigate a predator-prey model incorporating the Allee effect within a two-dimensional nonlinear reaction-diffusion framework, with the aim of understanding how local spatial refuges can promote the persistence of low-density populations by enabling them to surpass recovery thresholds. We first simulate an extinction-prone scenario in which initial densities fall below the Allee threshold, demonstrating that most populations tend toward extinction. We then introduce protected areas together with positive growth terms to facilitate survival. To assess the role of diffusion-reaction dynamics, we construct an objective function based on the shape and location of protected areas, and employ a bi-objective optimization approach. Our results reveal that as the weights of the objective function vary, the optimal protectedarea configuration shifts between fragmented and contiguous patterns. We begin with a single-species prey analysis and subsequently extend the model to include predators, where we use mathematical analysis to investigate the steady states of the two-species system.